A392484 Composite numbers k such that binomial(11*k, k) == 11^k (mod k).

10, 20, 55, 66, 5041, 45023, 20809897, 526896271, 12474611809

Offset

1,1

Comments

If k is prime, then binomial(m*k, k) == m^k (mod k) as a consequence of Fermat's little theorem and Babbage's congruence.

611384544067 and 4583656004721959 are terms.

If p is a prime such that p^2 divides 11^(p-1) - 1, then p^2 is a term. See for instance p = 71, p^2 = 5041.

Links

Table of n, a(n) for n=1..9.

Mathematica

okQ[k_]:=!PrimeQ[k]&&Mod[Binomial[11k, k], k]==PowerMod[11, k, k]&&k>3; Select[Range[6000], okQ] (* James C. McMahon, Jan 19 2026 *)

Prog

(Python)
from itertools import count, islice
from sympy import isprime
from oeis_sequences.OEISsequences import binom_mod
def A392484_gen(startvalue=4): # generator of terms >= startvalue
    for j in count(max(startvalue, 4)):
        if not isprime(j) and binom_mod(11*j, j, j) == pow(11, j, j):
            yield j
A392484_list = list(islice(A392484_gen(), 6))

Crossrefs

Cf. A002808, A084699, A080469, A109760, A109769.

Sequence in context: A254030 A188896 A067192 * A030004 A271512 A294473

Adjacent sequences: A392481 A392482 A392483 * A392485 A392486 A392487

Keyword

nonn,hard,more

Author

Chai Wah Wu, Jan 13 2026

Extensions

a(9) from Chai Wah Wu, Jan 25 2026

Status

approved